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A THEOP.EM ON UNIFOP7l~ PROVABILITY OF SCHEMES (f9fc•J, <f \\--92

Jpn Krajíèek

For ~arithmetic PA

en:v of the usuel schemes axiometizing peano

the collectionthe induction scheme I,- e.p;.

the least number princilJle L or the pigeonholescheme B,,

the f'ollowing holds (over some f'inite partDrinciDle PRF -

cf PA) :

cenotes the theory axiomatized by the inst8n-\A'here

ces cf

whether for 8~'r ~~hpmp ho1ds~

the methcds cf' R.L.Vauf!.ht [4)As G.Kreisel remerked,

p counterexample.e construction ofsurgest

Exemule 1

Define the scheme 11

:=f'R(x ,y) is a partial truth defin.i tion setis-

'fying Tarski's conditions for ~t-fOrmt:Ilas".

cf'This cen be obviousl '1 VTi t ten lover some fini te part T. \: o

PA) BS p single formule. Novl define the scheme S§ :

J .Paris 8sked '~Thether this is alVl'BYs the cp-se, i. e.,

.-." ~- [> 8xiomatizing PA <*) -' .

B~~ the usuAl dipgonpl ~rf!,ument. for itholas:

Thus:

i.e.

On the other sice there cennot be c setisfying the f'irst

D~rt cf (*) for pIl n; t~ke n:=c. it v:ould hold:J..e.

I l3c r- P'2C '

but PS there is (provpbly in I~2C)

fini tion v:e v,'ould elso hpve:

enc sO:

v'hich contred icts the knov'n fects.

Dne can psk then for v'hich schemes the property

a sufficient condition is that 12(tf:> holds. Obviously,

pnc I mutuelly prove Dne another in e "schemEltic ",TBY"

f111 usual schemes pxiomptizing PAt.This is the case Df

trn t~e nreFence cf the full pcheme of

x=~ -+ A(x)=A(y) , see Exemple 2.identity:

(Anelogical construction-define the scheme <Ø by

V t(-a(R, t) -') Con (I"t:' l+f ltJ )) - shows thp.t for no recursi ve

reetrictions but v'e f'hell omit this. Also schemes should

be forrnulated v,'ith metes;\7rnbols for variables to ret 811

elDhAbeticAl vBrients of the instAnces but \!Je shell omit

such ceteils here, cf. r21 .'

Fxiomatized by e fini-A scl:lem~tic sYf;tem is f' theor~r

te number of'schemes. We shell essume thet ell systems uncer

considerBtion hBve PS pn uncerlying logicel cplculus some

fixed Hilbert-style formuletion of the predicate cp:lculus

Vle shpll need f'ome notion of' of' ~ f'or-a coml)leyit~r

mulF... As the result is quite robust v!.r.t. 2 definition cf'

the complexity, the r'eeder c~n chaose his fp.vourite one

the quantif'ier coffiDlexity Dr, in the cp-the lofícal depth,

se r)(:=t!1e lanpuaf!e of' PA, the complexitJ, in terms of the

eri thmeticpl. hierBrchy. ,Let di' (A) denote an:v Dne of tne

fA: For ~ny schem8tic system there is a

constent cIA> O such that if d is an ~ -

proof v:ith k steps then there exists a se-*- ~

"'---~-~ d of o'•. -formulas= _B1' . . . ,Bknuence

sptisf'~inf:

li )

gi ven by 8 t, sui table ti

substitution is en ~ -proof. oThe term "suitpble" substitution meens that the substitution

~voids e clesh cf' vf1rif1bles - cf'. [2J. The sequence d-is

sbcve ccmplexities cf A.

(ii) c is e SUb~,~itution instance of d ~

( ". . ) .. ~ ,:j ~1.1.1. eny.t J.p~ t en~ e. DL: \.:

cl'lled A Droof'~echeme.

Fpct 1, if' d is an

t~en ther'e is e

(suitp.ble

Ak=A, cf

substitution-instpr:lce d'=Al,..,Ak'

Jk .the proof"-scherne d gl "'en by Ff\ct 1

such thpt

o

§2. The st.pte;:nent

For fA a schem8tic F'ystem let lA~k denote the set of'

in.c:tpnces of fA f'or JI!:.-f:ormUles of' the comnlexity pt most

k. I.e. if' the ccTnDlexity is def'ined in the terms cf' the

pri thmeticel hierprch;\T then I f' k = I ~ k . )

18Theorem: Let be t~'o schematic systems. Then the

li)

(iii)«there is D finite number of proof-schemes dl'..

..,d~ such that for eny ~-formula A there is

such thet

Prací': I

(ii)~(i) is triviel.

(i)~(iii): From Fect 1 it f'olloyrs thet f'or pny A -:proof'

d cf' an lB -instr;nce yrith c steps there is a proof'-

f'ollov,rinp three cond itions e,re equi valent:

3 c, fA t-c- lB /

i ~ r and a tI sui teblet1 substi tution é

*'scheme c O h,... Q , -.,lt C .,tLI of ~tenstile

are ere l'c 0 ""11 '"L. ,- .' i;;ite number

a• t:'sf'.~ntioll:v

~ cIA' c. Obviousl:,\'

such DToof'-sche11ef' vrtjich rre diff'er'enT ,. .

:11 f' :fo f ' r """"' F r. -.I... .- t 2 v'ith

iL1Fl r"UTf.berc := CIA mp o: "' te n s 'r,, ..., s

~:ne (Ji) . O

CM) ,e.f",.:

i:nDlieE the corlcitions af t'-le theor'em. As Eynected, the pn-

sv,'er is nep-pti,\re.

Exf',m-ole 2

F t I..' 1 ,., 'Ii t ' -f' 1 +' ft ' or, !ll.S e}':2:n~} e c -,nf'l.1. E'r:"lE .. ormu F..1 on o ('.e Dre-

cicpte cplculus v'ithout tr:e . f"ull scr.emeof identity

b:\~cec p f'ir. i te r}u~ber of

Let ~ h e .. )-" l n""" u "'!:'-eo< ~' l - c; (""O. '"" '-..

ifer:.tity-pxioms

of PA, L be. ~

•~,e S"Atem PX1O-

pnc r be PA Dlus the u~uel Rcherrle of indurtion:

Obviously (~~)t )holcs for L,I:

Assurr.e nov' the't the conditions of tne theorem/hold for L,.r:

!\T Fturp 1 •}tjte f' t i on rrif' ef' v'he t:'1er conc i t i on li~'<:e

As incependently obe.erved by D.Richf'rd e,On Bnd T .Yukemi:

j c,' ~m) n ,(tt I~ (m+n)(C) .

]c"'\iJ1, L ~ j >:, x+x=s(2rI1'1C)Cttti!T1T:'1:v that

for' ctsome n. A conty"rc ar!

(f) coes notA ~' 11~ B r,"' z hnco ob COer ' \ l eC~...,_J:ll.cc~..c"" " tr;.e ;'}oint "'fh.v

hald is t"•lpt in t\:e uRuel èe!..i'!~tion af I from L one uses

t he f'ull f:(~~eme of. 'I1. c. e r. t Ir: :)8rticular. D +' 0 1 , 0 ",' 1.'

n O'

\: ~ ~,. F,

t",.o f':neci~l Cf1ses of' PY'e nee"cel

Pl

P2

x=c ~ A(Y) -==. A(O) , fJr:c

i

(;[;1'.;J c ,L 10It is not known whether or

;]C,L tc P2 but by the example above at most one

of these statements may hold.

:;ICtr ~ </11It is easy to see that (apply induct-

) but it is notion to the formula (x=O -7 A(x) 'EA(O»)..,..

known whe ther "3 c ,I 1-0- 92 2 .

/AIf the lpnpUFtfe of is pllo,"ec~ to be A nroper

/8 P vl'riet:v oj' F.imilf'rextension of the lanpue,c-e of

is the ~receceE~or function.

in the lrcn•;u8peto be Rome f'inite pxiomptizption of ACAo

.,!I•' plus set vpripbles. Then the sa:ne erf:":ument v\Torks-cf'

[3,§6.] .

Ref~ren~es

[1) !I~. Bp8Z: Generelizing proofs v,'ith or'c er-lncuction,- -

!Ilpnuscrint.

to anpear

[3J c.T .Kre,jièek, P .PucllÉk: The nurnber of' Droof' lines 8nó' the

size cf' proof's in f'irst orcer lo~ic, Archive

,PP.69-84

[4J R.L.Vp,upht: Axiomatizebility by a scheme, J.Symbolic

Logic,

MpthemeticAl Institute

Žitna 25, PrehE". 1

115 67, Czechos1ovekie

in AnnBIs of' Pure Bnd Applied Lopic